SMALL SIGNAL ROTOR ANGLE STABILITY

1539pk
SMALL SIGNAL
(ROTOR ANGLE) STABILITY
Copyright © P. Kundur
This material should not be used without the author's consent

1539pk
SSS - 1
Small Signal (Rotor Angle) Stability
 Description of Small Signal Stability Problems
 local problems
 global problems
 Methods of analysis
 time-domain analysis and its limitations
 modal analysis using linearized model
 Characteristics of local plant mode oscillations
 Characteristics of interarea oscillations
 Enhancement of Small Signal Stability
 August 10, 1996 Disturbance of North American
Western Interconnected system
Outline

1539pk
SSS - 2
Small-Signal (Angle) Stability
 Small-Signal (or Small Disturbance) Stability is the
ability of a power system to maintain synchronism
when subjected to small disturbances
 such disturbances occur continually on the system
due to small variations in loads and generation
 disturbance considered sufficiently small if
linearization of system equations is permissible for
analysis
 Corresponds to Liapunov's first method of stability
analysis
 Small-signal analysis using powerful linear analysis
techniques provides valuable information about the
inherent dynamic characteristics of the power system
and assists in its robust design

1539pk
SSS - 3
 Instability that may result can be of two forms:
 aperiodic increase in rotor angle due to lack of
sufficient synchronizing torque
 rotor oscillations of increasing amplitude due to lack
of sufficient damping torque
 In today's practical systems, small signal stability is
usually one of insufficient damping of system
oscillations.
 local problems or global problems
 Local problems involve a small part of the system.
They may be associated with
 rotor angle modes
 local plant modes
 inter-machine modes
 control modes
 torsional modes
 Global problems have widespread effects. They are
associated with
 interarea oscillations

1539pk
SSS - 4
Local Rotor Angle Stability Problems
 Associated with either local plant mode oscillations or
inter-machine oscillations
 frequency of oscillation in the range of 0.7 to 2.0 Hz
 Local plant mode oscillations
 oscillation of a single generator or plant against rest
of the power system
 Inter-machine or inter-plant mode oscillations
 oscillation between the rotors of a few generators
close to each other
 Stability of the local plant mode oscillations is
determined by
 strength of transmission as seen by plant
 excitation control
 plant output and voltage
 Instability may also be associated with a non-oscillatory
mode
 encountered with manual excitation control

1539pk
SSS - 5
Global Rotor Angle Stability Problems
 A very low frequency mode involving all the generators
in the system
 system is essentially split into two parts
 generators in one part swing against generators in
the other part
 frequency in the order of 0.1 to 0.3 Hz
 Higher frequency modes involving sub-group of
generators swinging against each other
 frequency typically in the range of 0.4 to 0.7 Hz
Large interconnected systems usually have two
distinct forms of interarea oscillations:

1539pk
SSS - 6
Examples of Inter-area Oscillation modes
 The 0.28 Hz mode
in WECC
 The 0.49 Hz mode
in Eastern
US/Canada
interconnection

1539pk
Methods of Small-Signal
Stability Analysis

1539pk
SSS - 8
State Space Representation of a
Dynamic System
 The behaviour of a dynamic system can be described
by a set of first order differential equations in the
state-space form
 x is an n-dimensional state vector
 f is an n-dimensional nonlinear function
 u is a r-dimensional input vector
 The outputs of the system are nonlinear functions of the
state and input vectors
 y is an m-dimensional output vector
 g is an m-dimensional nonlinear function
 At steady state, the system is at an equilibrium point x0
satisfying
 uxfx ,
 uxgy ,
  0, 00 uxf

1539pk
SSS - 9
Linearization
 Nonlinear dynamic behaviour:
Output variables:
 At equilibrium point (x0,u0):
 Small perturbation about equilibrium point:
 New state equation:
 Since perturbations are small:
 f(x,u) can be expressed in terms of Taylor's series
expansion
 terms involving second and higher order powers of Dx
and Du may be neglected
 uxfx ,
 uxgy ,
  0, 000  uxfx
uuuxxx DD 00 ,
    uu,xxf
xxx
00
0
DD
D 

1539pk
SSS - 10
 A, B, C, D are the Jacobians of the system. A is also
referred to as the state matrix or the plant matrix.

























n
n
n
1
1
n
1
1
x
x
x
f
x
f
x
f
A 




uDxCy
uBxAx
DDD
DDD
Linearization cont'd

























n
n
n
1
1
n
1
1
u
x
u
f
u
f
u
f
B 





























n
m
n
1
1
m
1
1
x
g
x
g
x
g
x
g
C 





























r
m
n
1
1
m
1
1
u
g
u
g
u
g
u
g
D 





1539pk
SSS - 11
Analysis of Stability in the Small
(Small Signal Stability)
 The stability in the small of a nonlinear system is given
by the roots of the characteristic equation of the system
of first approximation, i.e., by the eigenvalues of the
state matrix A
 If the eigenvalues have negative real parts, then the
original system is asymptotically stable
 When at least one of the eigenvalues has a positive real
part, the original system is unstable
The theoretical foundation for the analysis of stability
in the small is based on Liapunov's first method:

1539pk
SSS - 12
Limitations of Nonlinear Time Domain
Analysis
 Results can be deceptive
 critical mode may not be sufficiently excited by the
chosen disturbance
 poorly damped modes may not be dominant in the
observed response
 It may be necessary to carry out simulations up to 20
seconds
 computational burden is high
 massive amount of data has to be analyzed
 This approach does not give insight into the nature of
the problem
 difficult to identify sources of the problem
 mode shapes not clearly identified
 corrective measures are not readily indicated
Using nonlinear time domain simulations to analyze
small signal stability problems has the following
limitations:

1539pk
SSS - 13
Post-Processing of Time-Domain Signals
Using Prony Analysis
 This is a method of spectral analysis
 a method of fitting a series of damped sinusoids to a
given signal
 For a signal record y(t), the Prony method fits a
function of the form
Ai the amplitude of the ith mode in the signal
the frequency (Hz) of the ith mode
the damping ratio of the ith mode
i the phase (rad) of the ith mode in the signal
 Prony analysis has the advantage that it can be used
on any signal: measured or simulated
 It can be used together with time-domain simulation to
estimate the frequency and damping of modes of
oscillations
 First published in 1795
 

M
1i
ii
t
i )tcos(eA)t(y i



2
f i
i 
22
i






1539pk
SSS - 14
Pros and Cons of the Prony Analysis
 A linear model can be aggregated from a time domain
response
 This gives useful information which can help verify
and complement the results of the linear modal
analysis
 However, the basic limitations of the time domain
analysis approach still remain
 The spectral analysis is subject to these drawbacks:
 it can evaluate only those modes that exist in the
variables monitored
 limited or no information about mode shapes
 it only looks at the system through a small window
 resolution of close modes presents challenges
 for a nonlinear system, different windows may give
different results
 Often, useful in extracting modal information from
measured time domain response

1539pk
SSS - 15
Modal Analysis Approach
 Modal analysis using eigenvalue approach has proven
to be the most practical way to analyze small signal
stability problems
 Advantages are:
 individual modes of oscillations are clearly identified
 relationships between modes and system
variables/parameters can be easily determined by
computing eigenvectors
 Frequency response, poles, zeros, and residues can be
easily computed. Such information is useful in control
system design

1539pk
SSS - 16
Eigenproperties of the State Matrix
 Eigenvalues and eigenvectors
 l is an eigenvalue
 f is the right eigenvector associated with l
  is the left eigenvector associated with l
 Modal matrices
  is the right eigenvector matrix
 Y is the left eigenvector matrix
 Relationships
 I is the unit matrix
  is a diagonal matrix: =diag[l1... ln]
ψψA
φAφ
l
l
 
 TT
n
T
2
T
1
n21
ψψψΨ
φφφΦ




ΛAΦΦ
IΨΦΦΛAΦ


1

1539pk
SSS - 17
Free Motion of a Linear Dynamic System
 Free motion of a linear dynamic system is described
by
 In order to eliminate the cross coupling between the
state variables consider the state transformation
 State space equations in z is a set of decoupled
differential equations
 The above represents uncoupled first order (scalar)
differential equations
 Time domain response
where is the initial condition
Axx 
zΦx 
zzΦAΦz 1
 
n,...,2,1izz iii  l
    t
ii
i
e0ztz l

   0xΨ0z ii 

1539pk
SSS - 18
Time Response of System Variables
 Response in terms of the original state vector:
x(t) = Φz(t)
 The time response of the state variable xi is given by
 a linear combination of n dynamic modes
corresponding to the n eigenvalues of the state matrix
 ci=ψix(0) represents the magnitude of the excitation of
the ith mode due to the initial conditions
 if the initial condition lies along the jth eigenvector, only
the jth mode will be excited (since ΨiΦj=0 for all i ≠ j)
 If the vector representing the initial condition is not an
eigenvector, it can be represented by a linear
combination of the n eigenvectors. The response of the
system will be the sum of n responses
 if a component along an eigenvector of the initial
conditions is zero, the corresponding mode will not be
excited.
  t
nin
t
22i
t
11ii
n21 ececectx lll
fff 

1539pk
SSS - 19
Eigenvalue and Stability
 A real eigenvalue corresponds to a non-oscillatory
mode
 A pair of complex eigenvalues l    j  correspond to
an oscillatory mode
 frequency of the mode
 damping ratio of the mode


2
f 
22






Eigenvalue plots and corresponding
time response

1539pk
SSS - 20
Damping Ratio
 Determines rate of decay of the amplitude of
oscillation
 For an oscillatory mode represented by a complex pair
of eigenvalues   j, the damping ratio is given by:
 The time constant of amplitude decay is
 The amplitude decays to 1/e or 37% of the initial
amplitude in t seconds or in cycles of oscillation
 As we are dealing with oscillatory modes having a
wide range of frequencies, damping ratio rather than
time constant is more appropriate for expressing the
degree of damping
 A 5 sec time constant represents amplitude decay to
37% of initial value in 5 cycles for 1 Hz local plant
mode, and in one-half cycle for 0.1 Hz interarea mode
 A damping ratio of 0.03 represents the same degree of
amplitude decay in 5 cycles for all modes
22







t
1

2
1

1539pk
SSS - 21
Right Eigenvector and Mode Shape
 A state variable is related to individual modes by
 fji is the ith element in the right eigenvector fj
 if fji = 0, the jth mode is unobservable in xi
 if fji is large, the jth mode will show up strongly in
xi
 therefore, fj determines the mode shape of the jth
mode
 The mode shape indicates the relative activities of the
state variables when a particular mode is excited.
 The magnitudes of the the eigenvector elements give the
extent of the activities; the angles of the elements give
phase displacements.
 The relative phase angles of the elements in the right
eigenvector can be used to determine the direction of
the oscillation in the associated state variables.
         tztztztztx nnijji2i21i1i ffff 

1539pk
SSS - 22
Methods to Show Mode Shape
 Mode shape geographical plot
This plot shows the generator rotor angles observable in
the mode at 0.28 Hz and 1.46% damping. The ones with
red cross oscillate against those with blue circles.
This is a mode in which units in northern WECC swing
against those in the southern WECC.

1539pk
SSS - 23
Left Eigenvector and Participation Factors
 A mode is related to individual state variables by
 ji is the ith element in the left eigenvector j
 if ji = 0, the jth mode cannot be controlled by xi
 if ji is large, the jth mode is largely determined by xi
 One problem in using directly the left eigenvector to
quantitatively determine the contribution of a state
variable to a mode is that the elements of the left
eigenvector are dependent on units and scaling
 The solution is to weight the left eigenvector by the right
eigenvector to obtain a quantity independent of unit and
scaling
 this is called the participation factor
 participation factors measure the participation of state
variables in modes. For instance, the larger pji the
more the state variable xi participates in the jth mode
jijijip f
         txtxtxtxtz njniji22j11jj  

1539pk
SSS - 24
Concept of Mode
 In frequency domain, a mode refers to a real eigenvalue
or a pair of conjugate complex eigenvalues
 Modal analysis of the linearized system model is the
perfect tool to obtain characteristics of individual
modes
 In time-domain, a mode is a component in a time
response that has a single frequency and damping,
together with other attributes of the sinusoid (amplitude
and phase angle).
 Prony analysis is one way to decompose time-domain
signals to obtain individual modes, although it is often
difficult to obtain the complete modal characteristics
 By its dynamic nature, a power system inherently
consists of many modes.
 If a mode has poor damping and when it is excited by a
disturbance, it can be observed from the post-
disturbance time-domain responses of variables that
have high observability in this mode
 When the system is unstable
 individual mode(s) unstable

1539pk
SSS - 25
Assessment of Modes Using Prony
Signal composed
from modes
identified
Simulated or
measured signal
With Mode #1
only (DC offset)
With Modes #1
and #2
With Modes #1,
#2, and #3
Mode
#
Amplitude
Frequency
(Hz)
Damping
(%)
Phase
()
1 29.9 0.0 N/A 0.0
2 7.5 0.72 4.3 36.2
3 6.2 0.67 6.2 67.6
Eigenvalue for mode #2: -0.19+j4.52
Eigenvalue for mode #3: -0.26+j4.21

1539pk
SSS - 26
Fig. 15.3 Rotor natural frequencies and mode shapes of a 555
MVA, 3,600 RPM steam-turbine generator

1539pk
SSS - 28
Controllability and Observability
 For a linear dynamic system
 Apply state transformation x = z,
 If the ith row of matrix -1B is zero, the ith mode is said
to be uncontrollable
 If the ith column of matrix C is zero, the ith mode is said
to be unobservable
 These concepts are useful in control system design
 First proposed by R.E. Kalman in 1960
DuCxy
BuAxx


DuzCΦy
BuΦΛzz

 1

1539pk
SSS - 29
Transfer Functions
 For a single-input-single-output (SISO) system
(assuming D=0)
 The transfer function is given by
 S1, S2, ... are poles (eigenvalues)
 z1. z2... are zeros
 R1, R2, ... are residues, given by:
cx
bAxx


y
u
   
 
 
    
    
n
n
n
n
s
R
s
R
s
R
sss
zszszs
K
s
su
sy
sG
l

l

l

lll







2
2
1
1
21
21
1
bAlc
bψcφR iii 

1539pk
SSS - 30
Computation of Eigenvalues
 The key problem in modal analysis is to compute
eigenvalues of the linearized model of power systems
 Depending on the system size or analysis objectives,
one of two computation options can be used:
a) Computation of all eigenvalues of the system using
QR method
 This is possible for small to medium sized power
systems (up to a couple of thousand states)
 All modes present in the model will be captured
b) Computation of partial eigenvalues in specific
frequency ranges using Arnoldi-type algorithm
 Applicable to systems of any size
 Ideal for analysis of interarea modes
 Selective computation also helps focus on the
modes of interest and facilitates analysis

1539pk
SSS - 31
Comparison of Computation Speed
 On one 2.0 GHz Pentium 4 PC with 512 MB memory,
running Windows 2000
 For a power system model with 34,381 buses, 3,870
generators, and 41,382 dynamic states
 Computation time is scalable to the system size and
number of contingencies to be processed
The following shows the comparison of typical
computation speed for time-domain simulations and
eigenvalue calculations:
Computation Time (min)
One 10-second time-domain simulation - TSAT 7.9
Mode scan in the frequency range of 0.2 and 0.8
Hz and damping range of 0% and 10% (24
modes were computed) - SSAT
2.9

1539pk
Local Plant Mode Oscillations

1539pk
SSS - 33
Characteristics of Local Plant Mode
Oscillations
 Local mode oscillation problems most commonly
encountered
 dates back to the 1950s and 1960s
 associated with units of a plant swinging against rest of
system
 Characteristics well understood
 analysis using block diagram approach (K-constants)
gives physical insight
 Encountered by a plant with high output feeding into weak
transmission network (K5 negative)
 more pronounced with high response exciters/AVR
 Adequate damping readily achieved using Power System
Stabilizers (PSS)
 excitation control

1539pk
SSS - 34
 First published by Heffron and Phillips to analyze a
single machine (or a plant) connected to a large
system (represented by an infinite bus) through a
transmission network
 System is represented by a block diagram as shown in
Fig. SSS-1 with
 K constants as parameters
 Dd, D, D Y fd, DEfd as the state variables
 This approach is limited to analysis of single machine-
infinite bus systems with generator damper windings
neglected
 here we use it to complement the results of
eigenvalue analysis
 provides physical insight into the effects of
excitation control
Block Diagram Approach to the Analysis of
Local Mode Problems

1539pk
SSS - 35
Fig. SSS-1 Block diagram of a synchronous machine with
excitation control
δ = ROTOR ANGLE (rads) Gex = EXCITER TRANSFER FUNCTION
ω = ROTOR SPEED (p.u.) GPSS = PSS TRANSFER FUNCTION
Ψfd = FIELD FLUX LINKAGE M = INERTIA CONSTANT (2H)
Efd = FIELD VOLTAGE
Ef = TERMINAL VOLTAGE

1539pk
SSS - 36
Block Diagram Interpretation
The following expressions form the basis for the block
diagram
 Rotor acceleration
 Electrical torque
 Field flux linkage
 Terminal voltage
 
D
dD
DD
D
0
em
dt
d
TT
M
1
dt
d
   fd21e KKT DYdDD
 
3
3
4fdfd
sT1
K
KE

dDDDY
   fd65t KKE DYdDD

1539pk
SSS - 37
Illustrative Example
 Consider the following simple system
 Initial condition
P = 0.9, Q = 0.3, Et = 1.0
 Small signal and transient stability are studies with
 classical model
 constant field voltage
 static exciter with AVR
 static exciter with AVR and PSS

1539pk
SSS - 38
Small Signal Stability with Classical Model
(Example 12.2)
 Generator output: P = 0.9, Q = 0.3
 K – constants: K1 = 0.758, KD = 0
 State variables: D, Dd
 Eigenvalues
 Eigenvectors
 Participation matrix
 
39.6j0
Hz02.1f,039.6j0
2
n1
l
l

















35.6j71.0
0j00.1
35.6j71.0
0j00.1
21 φφ










4.6503.04.6503.0
4.6503.04.6503.0

1539pk
SSS - 39
Small Signal Stability with Classical Model
cont'd
 Block diagram representation
KS=K1=0.758 KD=0
(a)
• Constant flux
• Positive synchronizing torque coefficient
• Negligible damping

1539pk
SSS - 40
Small Signal Stability with Constant Field
Voltage (Example 12.3)
 K-constants:
K1=0.764, K2=0.865, K3=0.323, T3=2.36, K4=1.42
 State variables:
D, Dd, DYfd
 Eigenvalues:
l1,2 = -0.11 ± j 6.41 ( = 0.017, fn = 1.02 Hz)
l3 = -0.20 + j 0
 At 1.02 Hz
KS=K1+KS(DYfd)=0.7643-0.00172=0.7626 pu torque/rad
KD=KD(DYfd)=1.53 pu torque/pu speed change
 Participation matrix
321
fd0021940170940170
00101501015010
00101501015010
lll
DY
dD
D











...
...
...

1539pk
SSS - 41
Steady-State Stability Limit with Constant
Field Voltage
Generator Output:
P = 1.387, Q = 0.462, Et = 1.0,
K1 = 0.457, K2 = 0.760, K3 = 0.303, K4 = 1.57
Eigenvalues:
l1,2 = -0.23  j4.95 ( = 0.046, fn = 0.8 Hz)
l3 = +0.006 + j 0 (aperiodic instability)
The steady-state synchronizing torque:
Ks = K1 - K2K3K4
= -0.0014

1539pk
SSS - 43
System Stability with Constant Field
Voltage
 Field flux variations are caused only by feedback of
through K4
 this represents the demagnetizing effect of armature
reaction
 Constants K2 and K3 are always positive; K4 is usually
positive
 Due to the phase lag introduced by field circuit time
constant (T3), the effect of DYfd due to armature reaction
is to introduce
 negative synchronizing torque at low frequencies
 positive damping torque and a small negative
synchronizing torque component at typical oscillating
frequencies of 1 Hz
 System stability depends on the net synchronizing
torque
 reaches steady state stability limit when
K1 = K2K3K4

1539pk
SSS - 44
Small Signal Stability with AVR
(Example 12.4a)
Case 1: Generator output P=0.9, Q=0.3
 K-constants:
K5 = -0.146, K6 = 0.417
 Eigenvalues:
l1,2 = +0.5j 7.23 (= -0.07, fn = 1.15 Hz)
l3 = -20.2 j 0 oscillatory instability
l4 = -31.2 j 0
 At 1.15 Hz:
KS(AVR+AR) = 0.2115 pu torque/rad
KD(AVR+AR) = -7.06 pu torque/pu speed change
Case 2: Generator output P = 0.3, Q = 0.1
 K – constants:
K5 = 0.025, K6 = 0.54
 Eigenvalues:
l1,2 = -0.04j 6.15 ( =0.007, fn = 0.98 Hz)

1539pk
SSS - 46
Effect of AVR
 Effect of AVR depends on the values of K5 and K6
 K6 is always positive
 K5 can be positive or negative
 K5 positive
 for low Xe and low P0
 effect of AVR is to introduce
 negative synchronizing torque
 positive damping torque
 K5 negative
 for high Xe and high P0
 effect of AVR is to introduce
 positive synchronizing torque
 negative damping torque
 above effect is more pronounced with higher exciter
response
 Situations with K5 negative is commonly encountered
 this could lead to oscillatory instability
 an effective solution is to use PSS

1539pk
SSS - 47
Principle of PSS
 Uses auxiliary stabilizing signal to control excitation
 most logical signal is D
 If transfer functions of exciter and generator were pure
gains
 direct feedback of D would result in damping
torque
 In practice, the generator and possibly exciter exhibit
frequency dependent gain and phase characteristics
 phase compensation results in a pure damping
torque component
 Exact phase compensation results in a pure damping
torque component
 over compensation introduces negative
synchronizing torque component
 under compensation introduces positive
synchronizing torque component

1539pk
SSS - 48
Block Diagram of a synchronous machine with excitation
control

1539pk
SSS - 49
Fig. 12.14 Thyristor excitation system with AVR and PSS

1539pk
SSS - 50
Small Signal Stability with PSS
(Example 12.3b)
 Generator Output:
P = 0.9, Q = 0.3
 Eigenvalues:
= -1.0 ± j6.6 (ζ = 0.15, fn = 1.06 Hz)
= -19.8 ± j12.8
= -39.1 ± j0
= -0.74 ± j0
 At 1.06 Hz:
KS = K1+KS(AVR+AR) + KS(PSS)
= 0.7643 + 0.21 - 0.14
= 0.8293 pu torque/rad
KD = KD(AVR+AR) + KD(PSS)
= -8.69 + 22.77
= 14.08 pu torque/pu speed change
 represent rotor angle mode
represent "exciter" mode
2,1l
4,3l
2,1l
4,3l
5l
6l

1539pk
SSS - 51
Time Domain Responses of Different Models
Fig. SMIB-1 Rotor Angle

1539pk
SSS - 52
Power System Stabilizers
 Small signal stability problem is usually one of
insufficient damping of system oscillations
 Power system stabilizers (PSS) are the most cost
effective means of solving SSS problems
 The purpose is to add damping to the generator rotor
oscillations
 This is achieved by modulating the generator
excitation so as to develop a component of electrical
torque in phase with rotor speed deviations
 Common input signals include: shaft speed, integral of
power and generator terminal frequency

1539pk
SSS - 53
Delta-Omega Stabilizer
(Based on Shaft Speed Signal)
a) Application to Hydraulic Units:
 Used successfully since mid-1960s
 Requires minimization of noise
 noise below 5 Hz level must be <0.02%
 shaft runout (lateral movement) produces most
significant noise components
 low frequency noise cannot be removed by
conventional filters; elimination must be intrinsic in
method of signal measurement
 Speed outputs are summed from several locations on
shaft
 Stabilizer disconnected at gate positions below 70%
 prevent effects of turbine vibrations at partial gate
opening
b) Application to Thermal Units:
 Stabilizer may cause instability of torsional
oscillations
 Speed should be sensed at nodes of torsional modes;
requires torsional filters

1539pk
SSS - 54
Delta-Omega Stabilizer cont'd
 The disadvantages of the Delta-Omega stabilizer are:
 torsional filer is needed. This can introduce phase
lag at lower frequencies and destabilize exciter
mode
 it imposes maximum limit on stabilizer gain
 custom design is required for each unit to deal with
torsional modes
 Delta-P-Omega stabilizer was developed to overcome
these problems

1539pk
SSS - 55
Frequency Based Stabilizers
 The frequency signal is obtained in one of two ways
 terminal frequency signal is used directly as the
stabilizer input signal
 Vt and It are used to derive the frequency of a
voltage behind a simulated machine reactance so as
to approximate the machine rotor speed
 On steam-turbine units torsional modes must be
filtered
 Gain may be adjusted to obtain the best possible
performance under weak ac transmission system
conditions
 Better performance for damping interarea modes than
speed-based stabilizers
 Disadvantages
 spike may occur in EFD during a rapid transient
(terminal frequency signal will see a sudden phase
shift)
 frequency signal often contains power system noise
 torsional filtering is required

1539pk
SSS - 56
Delta-P-Omega Stabilizer
 Signal proportional to rotor speed deviation can be
derived from the accelerating power
 It is important to derive Deq which does not contain
torsional modes
 Torsional components are inherently attenuated in the
integral of DPe signal
 The problem is to measure the integral of DPm free of
torsional modes
 Neglecting DPm is unsatisfactory if mechanical power
changes
 Delta-P-Omega uses:
 dtPP
M
1
emeq DDD
em PMP DDD

1539pk
SSS - 57
 Pm changes are slow; so, simple low pass filter can be
used to remove torsionals
 Advantages over the Delta-Omega stabilizer
 DPe signal has a high degree of torsional attenuation
 generally there is no need for a torsional filter in the
main stabilizing path
 higher stabilizer gain is possible which results in
better damping of system oscillations
 Standard design for all units (end-of-shaft speed
sensing arrangement with a simple torsional filter)
Delta-P-Omega Stabilizer cont'd
         





D
D

D
D s
Ms
sP
sG
Ms
sP
s ee
eq

1539pk
SSS - 59
Interarea Oscillation Problems
 Used to occur in isolated situations
 since mid-1980s has become more commonplace
 increasingly being identified in planning and
operating studies
 In the early 1990s major efforts undertaken to
investigate interarea oscillation problems:
 Canadian Electric Association (CEA) Research
Project 294 T622 Report, 1993
 IEEE System Oscillations Working Report 95TP101,
1994
 CIGRE Technical Brochure on “Analysis and Control
of Power System Oscillations" prepared by
TF38.01.07, 1996
 We first illustrate the nature of interarea oscillations by
considering a simple two area system
 see paper #2 in Appendix

1539pk
SSS - 60
Example 12.6: A Simple Two Area System
 Two similar areas connected by a weak tie. Each area
consists of two 900 MVA thermal units, loaded to 700
MW
 With all 4 units on manual excitation control,
determine:
 all eigenvalues of the system state matrix
 for each mode, state variables with high
participation
 frequencies, damping ratios, and mode shapes of
rotor angle modes

1539pk
SSS - 61
Example 12.6 (cont'd)
Table E12.3 System modes with manual excitation control
Fig. E12.10 Mode shapes of rotor angle modes with manual
excitation control
(a) Inter-area mode
f=0.545 Hz, =0.032
(b) Area 1 local mode
f=1.087 Hz, =0.072
(c) Area 2 local mode
f=1.117 Hz, =0.072
G4
G3
G2
G1
G3
G4G2
G1
zero
eigen
values

1539pk
SSS - 62
Example 12.6 (cont'd)
 Determine the frequencies and damping ratios of rotor
angle modes with different types of control
 Local inter-machine modes have same degree of damping with
DC and thyristor exciters (with and without TGR)
 Inter-area mode has
 a small positive damping with DC exciter
 a small negative damping with a high gain thyristor
exciter
 a large negative damping with thyristor exciter with TGR
 PSS results in a significant damping of all the rotor angle
modes
Type of excitation
control
Eigenvalue/(frequency in Hz, damping ratio)
Inter-area mode Area 1 local mode Area 2 local mode
(i) DC exciter -0.018 ± j 3.27
(f = 0.52, ζ = 0.005)
-0.485 ± j 6.81
(f = 1.08, ζ = 0.07)
-0.500 ± j 7.00
(f = 1.11, ζ = 0.07)
(ii) Thyristor with
high gain
+0.013 ± j 3.84
(f = 0.61, ζ = -0.008)
-0.490 ± j 7.15
(f = 1.14, ζ = 0.07)
-0.496 ± j 7.35
(f =1.17, ζ = 0.07)
(iii) Thyristor with
TGR
+0.123 ± j 3.46
(f = 0.55, ζ = -0.036)
-0.450 ± j 6.86
(f = 1.09, ζ = 0.06)
-0.462 ± j 7.05
(f =1.12, ζ = 0.06)
(iv) Thyristor with
PSS
-0.501 ± j 3.77
(f = 0.60, ζ = 0.13)
-1.826 ± j 8.05
(f =1.28, ζ = 0.22)
-1.895 ± j 8.35
(f =1.33, ζ = 0.22)

1539pk
SSS - 63
Redundant State Variables and Zero
Eigenvalues
 Formulation of system state equations uses absolute
change in machine rotor angle and speed
 will result in one or two zero eigenvalues
 One of the zero eigenvalues associated with lack of
uniqueness of absolute rotor angle
 angles of all machines may be changed by same
value without affecting stability
 absent if "infinite buses" included
 Second zero eigenvalue exists if all generator toques
are independent of speed deviation
 no governors and KD = 0
 Zero eigenvalues may not be computed exactly due to
limited computational accuracy

1539pk
SSS - 64
Effect of Generator and Excitation System
Characteristics on Inter-Area Mode Shapes
Note: 1. These results are from Paper #2 in Appendix
2. Loads assumed to have constant Z characteristics

1539pk
SSS - 65
Case 2B: Generators Real Power

1539pk
SSS - 66
Case 2B: Generators Real Power and Loads

1539pk
SSS - 67
Characteristics of Low Frequency
Interarea Oscillations (LFIO)
 Oscillations between two groups of generators
 Two distinct forms:
a) A very low frequency mode involving all generators
 entire system split into two parts, with
generators in one part swinging against
generators in the other part
 frequency in the range: 0.1 to 0.3 Hz
b) Higher frequency modes involving a subgroup of
generators swinging against another subgroup
 frequency in the range: 0.4 to 0.7 Hz

1539pk
SSS - 68
Interarea Modes in the WSCC system

1539pk
SSS - 69
Incidents of Interarea Oscillations
Many incidents of poorly damped or unstable oscillations
have been reported worldwide:
 Michigan-Ontario-Quebec: 0.25 Hz in 1959
 Saskatchewan-Manitoba-Ontario West: 0.35-0.45 Hz in the
1960s
 WSCC (WECC): Between 0.1 - 0.33 Hz in 1967-1980; 0.28 Hz
August 1996 and August 2000
 MAPP: Over 70 unstable oscillations (0.12-0.25 Hz) in 1971 and
1972
 NPCC: 0.24-0.4 Hz in 1985
 Nordel: 0.5 Hz in late 1960; 0.33 Hz in 1980
 Italy-Yugoslavia-Austria: 0.17-0.22 Hz in 1971-1974
 Australia: 0.6 Hx in 1975; 0.2 Hz in 1982 and 1983
 Scotland-England: 0.5 Hz in 1978 and 1979
 Taiwan: 1.0-1.1 Hz in 1984
 Ghana-Ivory Coast: 0.6-0.7 Hz in 1985
 Southern Brazil: 0.5, 0.8, 1.2 Hz in 1985-1987
 Scandinavia: 0.5 Hz in January 1997

1539pk
SSS - 70
LFIO System Experiences:
Ontario Hydro/NPCC
 First observed in 1985 in planning and operating
studies
 Confirmed by on-line measurements
 Involved the entire NPCC and eastern interconnected
system
 Frequency varied between 0.25 and 0.4 Hz depending
on operating conditions and load levels
 Mode shape for one condition shown in Figure
 Based on extensive investigations, following remedial
measures taken by OH:
 returned PSS on all major units (static exciters)
 retrofitted Pickering NGS with PSS (AC exciters);
most effective since close to load centre
 PSS for new units at Darlington NGS designed to
damp LFIO
 monitors installed throughout Ontario

1539pk
SSS - 71
Shape of 0.2 Hz Mode

1539pk
SSS - 72
Effect of Excitation Control of Darlington
Units on the Interarea Mode
Type of Excitation Control
Interarea Mode
Frequency Damping Ratio
a) Thyristor Exciter with high
transient gain
0.192 Hz 0.009
b) Thyrister Exciter with
transient gain reduction
0.187 Hz -0.057
d) Thyristor Exciter with high
transient gain and PSS
0.179 Hz 0.122
Note: Based on results presented in Paper #1 in Appendix

1539pk
SSS - 73
CEA Research Project 294 T622
Conclusions regarding fundamental nature of LFIO:
 Characteristics (mode shape, damping) of LFIO are a
complex function of:
 network configuration/strength
 load characteristics
 types of excitation systems and their locations
 Load characteristics, in particular, have a major effect
 more pronounced with slow exciters
 In a stressed system, motor or constant power load at
 receiving end has adverse effect on damping
 sending end has slightly beneficial effect
 A mode of oscillation in one part of system can excite
units in a remote part due to mode coupling
 Analysis requires detailed and same level of
representation throughout the system

1539pk
SSS - 74
CEA Project 294 T622 cont'd
Damping of LFIO with PSS
 The controllability of LFIO with PSS is a function of:
 location of units with PSS
 characteristics and locations of loads
 types of exciters on other units
 Damping of LFIO wit PSS achieved primarily by
modulating loads
 Identification of units on which PSS most effective:
 a high participation factor is a necessary but not
sufficient condition
 initial screening by participation factors
 residues and frequency responses can supplement
screening

1539pk
SSS - 75
Enhancement of Small-Signal Stability
1. Excitation Control: Power System Stabilizers
2. Supplementary Control of HVDC Links, SVCs,
and other FACTS devices

1539pk
SSS - 76
Damping Ratio (ζ)
 An important index related to small-signal stability
 The nature of system response largely depends on
the damping ratios of individual modes
 The minimum acceptable level of damping below
which the power system cannot be operated
satisfactorily is not clearly established
 Situations with damping ratios of less than 0.02 for
local plant mode and interarea mode oscillations
must be accepted with caution
 In addition to the absolute value of damping ratio,
what is important is its sensitivity to variations in
operating conditions and system parameters
 A low damping ratio but less sensitive to operating
conditions and system parameters is often
acceptable
 In the design of power system stabilizer and other
forms of controllers for damping power system
oscillations, a good design target is to have a damping
ratio ζ of at least 0.1 for the mode(s) of interest

1539pk
SSS - 77
Excitation Control Design
Design objectives:
 Maximize the damping of the local plant modes as well
as interarea mode oscillations without compromising
the stability of other modes
 Enhance system transient stability
 Not adversely affect system performance during major
system upsets which cause large frequency
excursions
 Minimize the consequences of excitation system
malfunction due to component failures

1539pk
SSS - 78
Fig. 17.5 Block diagram of thyristor excitation system
with PSS

1539pk
SSS - 79
Design Considerations
 Exciter gain
 high value of KA for transient stability (200)
 transient gain reduction (TGR) is required only if
voltage regulator time constant is large or exciter
has significant time delays
 TA about 1 second
 TB about 10 seconds
 TGR not required for Thyristor exciters
 Phase lead compensation
 compensate for lag between exciter input and
resulting electrical torque
 design should provide damping over wide range of
frequency to cover local and interarea modes

1539pk
SSS - 80
Design Considerations cont'd
 Phase lead compensation (con't)
 compute the frequency response between the
exciter input and the generator electrical torque with
the generator speed and rotor angle remaining
constant (assuming large inertia for the machine)
 frequency response of any machine is sensitive to
the Thevenin equivalent system impedance at its
terminals but relatively independent of the dynamics
of other machines (assuming other machines are
infinite buses)
 resulting phase characteristic has a relatively simple
form free from the effects of natural frequencies of
the external machines
 select characteristic suitable for different system
conditions
 Do not overcompensate !

1539pk
SSS - 81
Stabilizing Signal Washout
 High pass filter which prevents steady change in
speed from modifying the field voltage
 Washout time constant TW should be high enough to
allow signals associated with oscillations in rotor
speed to pass unchanged
 From the viewpoint of washout function, can be in the
range of 1 to 10 seconds
 pass stabilizing signals at the frequencies of interest
 not so long that it leads to undesirable generator voltage
excursions as a result of stabilizer action during system
islanding conditions
 For local plant modes, TW of 1.5 s or higher
satisfactory
 TW of less than 5 s results in significant phase lead at
low frequencies associated with interarea oscillations
 this can reduce the synchronizing torque component
 For systems with dominant interarea oscillations
 either set TW to about 10 s, or
 use one of the phase compensation blocks to provide
phase lag at low frequencies

1539pk
SSS - 82
Stabilizer Gain
 The stabilizer gain, KSTAB, has an important effect on
damping or rotor oscillations
 It is necessary to examine the effect of KSTAB for a
wide range of values
 Damping increases with an increase in KSTAB up to a
point
 Gain is set to provide maximum damping with the
following considerations:
 Delta-Omega stabilizer: due to the effect of the
torsional filter, the stability of the "exciter mode"
becomes an overriding consideration
 Delta-P-Omega stabilizer: exciter mode stability is
not a problem, and a considerably higher value of
gain is acceptable – limited only by amplification of
signal noise considerations

1539pk
SSS - 83
Stabilizer Limits
 Positive output limit is set at a relatively large value
(0.1 to 0.2 pu)
 This allows a high level of contribution from PSS
during large swings
 Et limiter needed
 High gain limiter can cause torsional mode instability
(Et has small components of torsionals) ... choose TC
and TD to provide high attenuation at torsional
frequencies, in addition to ensuring adequate degree
of limiter loop stability
 Negative limit of -0.05 to -0.1 pu allows sufficient
control range and satisfactory transient response

1539pk
SSS - 84
Setting Verification
 Overall performance should be evaluated for the
stabilizer parameter setting
 small signal stability program can be used to
examine performance over range of conditions
 there should be no adverse interactions with the
controls of other nearby generating units and
devices, such as HVDC converters and SVCs
 transient stability and long-term stability simulations
should also be used to verify the performance
 coordination with other protections and controls,
such as Volts/Hz limiters and overexcitation/
underexcitation protection

1539pk
SSS - 85
Commissioning and Field Verification
 Actual response of unit with PSS measured and used
to verify analytical results
 step change to AVR reference
 disturbance external to plant, e.g., line switching
 If there are discrepancies between measured and
computed responses
 models modified and revised PSS settings
determined
 During commissioning, PSS gain increased slowly up
to twice the chosen setting
 exciter mode stability margin
 input signal noise amplification
 On-line tuning of PSS impractical !

1539pk
SSS - 86
Hardware Design Considerations
 Allow setting of PSS parameters over sufficiently wide
range
 Ensure high degree of functional reliability and
flexibility for maintenance
 component redundancy
 duplicate PSS, AVR
 Built-in facilities for dynamic tests
 routine testing to avoid undetected failures
 Importance of good design features often overlooked
 many instances of operators disconnecting PSS

1539pk
SSS - 87
Example of Stabilizer Application
 Two thermal 488 MVA units equipped with thyristor
excitation systems
 Units exhibit two dominant rotor oscillation modes: an
interarea mode of about 0.5 Hz and a local inter-
machine mode of about 2.0 Hz
 Objective of excitation control design is to enhance
the transient as well as small signal stability of the
power system
 Examine the performance using slow rotating exciters
also

1539pk
SSS - 88
Thyristor excitation system:
 High exciter gain of 212 (with no TGR) is used to
ensure good transient stability performance
Power system stabilizer

1539pk
SSS - 89
Slow rotating excitation system:
 Self-excited dc exciter
1. Type DC1A exciter model

1539pk
SSS - 90
Slow signal stability performance:
Selection of PSS parameters
 Determination of phase-lead compensation
 G1 and G2 are represented with large inertia - all
others as infinite buses
Type of exciter
Local inter-machine mode Inter-area mode
Frequency  Frequency 
(a) Thyristor (no PSS) 1.823 Hz 0.049 0.550 Hz 0.006
(b) Rotating exciter 1.793 Hz 0.075 0.498 Hz 0.046

1539pk
SSS - 91
 Parameters for compensation shown
T1 = 0.06 T2 = 0.02 T3 = 1.5 T4 = 4.0 TW = 7.5
 Other parameters
Vsmax = 0.2 Vsmin = -0.05 VLS = 1.15 KL = 17
TC = 0.025 TD = 1.212 TRL = 0.01
KSTAB
Local inter-machine mode Inter-area mode
Frequency  Frequency 
0 1.823 Hz 0.049 0.550 Hz 0.006
20 2.079 Hz 0.156 0.547 Hz 0.087
30 2.218 Hz 0.197 0.548 Hz 0.124
40 2.366 Hz 0.227 0.533 Hz 0.156

1539pk
SSS - 92
Fig. 17.13 Response of unit G1 to a five-cycle three-
phase fault; peak load conditions

1539pk
SSS - 93
Selection of PSS Location
 For damping inter-area oscillations, best locations for
PSS may not be obvious in large systems
 PSS adds damping to an inter-area mode largely by
modulating system loads
 PSS with regard to a local mode is only slightly
affected by the load characteristics
 Understanding these mechanisms is essential
 Participation factors corresponding to speed
deviations of generating units are very useful for initial
screening
 Rigorous evaluation using residues and frequency
responses should be carried out to determine
appropriate locations for the stabilizers

1539pk
SSS - 94
Supplementary Control of HVDC Links,
SVCs and FACTS
 Main tasks in controller design:
 Selection of device and its location
 option may not always exist
 based on participation factors, residues, and
frequency response
 Selection of feedback signal:
 modes of concern must be observable in signal(s)
 based on frequency response between device input
and potential signals
 Controller design procedure
 a variety of linear techniques available
 varying degrees of procedure automation,
complexity, robustness, and applicability to large
system
 Test overall performance
 broad range of conditions and contingencies
 small-signal and transient stability

1539pk
SSS - 95
Approach
 The physical mechanism by which non-generator
devices contribute to damping of inter-area
oscillations in highly meshed networks is not obvious
 Control design techniques based on physical
principles, such as in PSS design, cannot be readily
applied
 Control design methods based on linear control theory
were investigated

1539pk
SSS - 96
Controller Design Methods Investigated
 Phase and gain margin
 based on Nyquist criteria
 controller is designed to improve phase and gain
margins of the closed loop system
 Pole placement
 controller is designed so that the closed loop
system has a pole (eigenvalue) at a specific location
 H-infinity
 a computer aided control design technique
 minimize H-infinity norm of the system transfer
function from the disturbance to the output over the
set of all stabilizing controllers
 produces a controller that is robust in some sense
 reduced order system model required

1539pk
SSS - 97
Example of HVDC Link Modulation
 Design of controller for an embedded HVDC link (with
parallel ac paths)
 System based on the WSCC system with two HVDC
links:
 Pacific Intertie, and
 Intermountain
 Controller designed for Pacific DC Intertie to improve
damping of 0.3 Hz north-south oscillations
 All three methods of control design gave satisfactory
results

1539pk
SSS - 98
Table 1
Low Frequency Modes of Original System
Mode
No.
All ccts I/S Grizzly-Malin O/S
Freq. (Hz) Damp Ratio Freq. (Hz) Damp Ratio
1. 0.298 0.079 0.284 0.077
2. 0.446 0.059 0.442 0.058
3. 0.607 0.045 0.606 0.046
4. 0.735 0.011 0.724 0.006
5. 0.747 0.073 0.746 0.072
6. 0.779 0.048 0.781 0.053
7. 0.804 0.045 0.796 0.042
8. 0.862 0.015 0.861 0.015
Note: Active component of loads assumed to have constant
current characteristics

1539pk
SSS - 99
Selection of Modulating Signal
 Mode to be damped, must be
 observable in the signal
 controllable by the device
 Mode 1 (north-south mode) is controllable by the DC Intertie
Mode 4 (Arizona-California) is not controllable by the DC
Intertie but may be stabilized by PSS at Helms GS in California
 PSS location identified by participation factor
 Several signals considered for modulating the DC Intertie
 difference between angles of rectifier and inverter ac bus
voltages selected for modulating the rectifier controls

1539pk
SSS - 100
Frequency Response for Angle Difference in the Post
Disturbance System

1539pk
SSS - 101
Derivation of Reduced Order Model
 For application H-infinity robust controller design
technique
 Objective:
Reduce a large order system model with up to 20,000
states to a transfer function model of order less than
15 which captures the essential characteristics of the
system
 Approaches
 compute poles and zeros in a dynamically reduced
model and eliminate close poles/zeros
 compute important poles and zeros in large system
and supplement the transfer function using system
identification techniques

1539pk
SSS - 102
Reduced Transfer Function for H-infinity
Control Design
Full system: 3866 states
Reduced transfer function: 12th order

1539pk
SSS - 103
Frequency Response Between the Rectifier Current Reference and
Difference Between Rectifier and Inverter AC Voltage Phase Angles

1539pk
SSS - 104
Table 2
Effect of HVDC Modulation on Mode 1 with
Grizzly-Malin cct O/S
Controller
Design
None
Pole
Modification
Phase/Gain
Margin
H-infinity
Freq. (Hz) 0.284 0.294 0.316 0.311
Damp Ratio 0.077 0.169 0.173 0.176
Table 3
Effect of PSS at Helms on Mode 4 with G-M cct O/S
PSS Out-of-Service In-Service
Freq. (Hz) 0.724 0.730
Damp Ratio 0.006 0.016

1539pk
SSS - 105
Summary
Method
Allowable
System
Model
Effect on
Other Modes
Achievement of
Robustness
Handle on
Damping
Phase and
Gain Margins
Large
Can be
considered
Compromise in
design
Good
Pole Placement Large
Difficult to
predict
Compromise in
design
Very good
H-infinity
Reduced
(<15 states)
Considered
in design
Automatically
considered
Weak

1539pk
SSS - 106
Conclusions
General:
 Each of the three methods has its advantages and
disadvantages
 With care and judgment, any method can be applied
successfully
H-infinity:
 Robustness is built into the method
 Selection of the weighting functions is critical
 Method needs reformulation to be more directly
applicable to power systems

1539pk
SSS - 107
WSCC August 10, 1996 Disturbance
Analysis and Mitigation
Studies conducted by the Power System Studies Group
Powertech Labs Inc.
Vancouver, Canada

1539pk
SSS - 108
WSCC August 10, 1996 Disturbance Initial
Conditions
 High ambient temperatures in Northwest; high power
transfer from Canada to California
 Prior to main outage, three 500 kV line sections from
lower Columbia River to load centres in Oregon were
out of service due to tree faults
 Line outages caused voltage reduction in lower
Columbia River area from around 540 kV to 510 kV
 Main outage at 15:47:36, loss of Ross-Lexington 230 kV
line due to tree flash over
 Growing 0.234 Hz oscillations caused:
 tripping of lines resulting in formation of four islands
 loss of 30,500 MW load

1539pk
SSS - 109
Event 4: 15:47:36 Ross-Lexington 230 kV - flashed to tree (+ 115 kv cct loss)
Event 2: 14:52:37 John Day - Marion 500 kV LG - flashed to tree
Event 1: 14:06:39 Big Eddy - Ostrander 500kV LG fault - flashed to tree
Event 3: 15:42:03 Keeler-Alliston 500 kV - LG - flashed to tree
Event 5: 15:47:36-15:48:09 8 McNary Units trip
2
1
3
4
4 5

1539pk
SSS - 110
Recordings showed undamped oscillations throughout
WSCC with a frequency of about 0.23 Hz

1539pk
SSS - 111
WSCC August 10, 1996 Disturbance cont'd
As a result of the
undamped
oscillations, the
system split into
four large islands
Over 7.5 million
customers experienced
outages ranging from a
few minutes to nine
hours! Total load loss
30,500 MW

1539pk
SSS - 112
Replication of the Event in Time Domain
MEASURED
RESPONSE
SIMULATED
RESPONSE

1539pk
SSS - 113
Eigenvalue Scan on Simulation Model

1539pk
SSS - 114
Application of Power System Stabilizers
Mode shape and participation
factors were computed for the
critical mode
Participations of generator
speed terms, controllability
and observability used to find
best locations for PSS tuning
or additions

1539pk
SSS - 115
Sites Selected for PSS Modifications
San Onofre
(Addition) Palo Verde
(Tune existing)

1539pk
SSS - 116
Verification of PSS Effectiveness
With existing controls
Eigenvalue = 0.0597 + j 1.771
Frequency = 0.2818 Hz
Damping ratio = -0.0337
With PSS modifications
Eigenvalue = -0.0717 + j 1.673
Frequency = 0.2664
Damping ratio = 0.0429

1539pk
SSS - 117
Design of HVDC Modulations
 HVDC shown (as expected) to have low participation in
mode
 Often however, HVDC can be modulated to improve
damping provided adequate input signal is found and
proper compensator is designed
 Frequency responses were examined for several
potential input signals
 Frequency response magnitude identifies local bus
frequency as having good observability/
controllability of mode of interest
 Frequency response phase used to design
compensator which provides proper modulation signal
to HVDC controls
 Time-domain and eigenvalue analysis used to verify
modulation performance

1539pk
SSS - 118
Verification of Effectiveness of HVDC Modulation
Simulation event
Without HVDC Modulation
Eigenvalue = 0.0597 + j
1.771
Damping ratio = -0.0337
With HVDC Modulations
Eigenvalue = -0.108 + j
1.797
Damping ratio = 0.0602

1539pk
SSS - 119
Appendix to Section on
Small-Signal Stability
Copies of Papers:
1. Application of Power System Stabilizers for
Enhancement of Overall System Stability
2. A Fundamental Study of Inter-Area Oscillations in
Power Systems
3. Analytical Investigation of Factors Influencing Power
System Stabilizers Performance
4. Effective Use of Power system Stabilizers for
Enhancement of Power system Reliability

SMALL SIGNAL ROTOR ANGLE STABILITY

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to SMALL SIGNAL ROTOR ANGLE STABILITY

Similar to SMALL SIGNAL ROTOR ANGLE STABILITY (20)

More from Power System Operation

More from Power System Operation (20)

Recently uploaded

Recently uploaded (20)

SMALL SIGNAL ROTOR ANGLE STABILITY